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Chapter 5: Problem 11

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{11 \pi}{12}=\frac{3 \pi}{4}+\frac{\pi}{6}$$

Short Answer

Expert verified
The exact values for sine, cosine and tangent of the angle \(\frac{11 \pi}{12}\) are \(\frac{\sqrt{6}-\sqrt{2}}{4}\), \(\frac{-\sqrt{6}-\sqrt{2}}{4}\) and -1 respectively.

Step by step solution

01

Find the Values for the Two Angles

First, convert \(\frac{3 \pi}{4}\) and \(\frac{\pi}{6}\) from radians to degrees. Here, \(\frac{3 \pi}{4}\) translates to 135 degrees and \(\frac{\pi}{6}\) translates to 30 degrees.
02

Find the Sine, Cosine, and Tangent for Both Angles

Use the known values of sine, cosine and tangent for the angles of 135 degrees and 30 degrees. For \(\frac{3 \pi}{4}\) or 135 degrees, \(\sin(135^o)=\frac{\sqrt{2}}{2}\), \(\cos(135^o)=-\frac{\sqrt{2}}{2}\), \(\tan(135^o)=-1\). For \(\frac{\pi}{6}\) or 30 degrees, \(\sin(30^o)=\frac{1}{2}\), \(\cos(30^o)=\frac{\sqrt{3}}{2}\), \(\tan(30^o)=\frac{\sqrt{3}}{3}\).
03

Apply the Sum of Angles Formulas

Finally, apply the sum of angles formulas for sine, cosine, and tangent: \[\sin(a+b)=\sin a \cos b + \cos a \sin b\], \[\cos(a+b)=\cos a \cos b - \sin a \sin b\], and \[\tan(a+b)=\frac{\tan a + \tan b}{1-\tan a \tan b}\]. Here, \(a\) is \(\frac{3 \pi}{4}\) and \(b\) is \(\frac{\pi}{6}\). By substituting exact values of sine, cosine, and tangent of both the angles, find the sine, cosine, and tangent of \(\frac{11 \pi}{12}\).

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